Fourier Series of modulus[t] - example of this?

• ZedCar
In summary, a user is asking for a worked example of the Fourier Series for a given function. Another user provides a link to an example and explains the difference between the function given and the one shown in the example. They also discuss the calculation of coefficients for the Fourier series and clarify an equation written in the original statement.
ZedCar

Homework Statement

Does anyone know of a website, or a book, where I can see a worked example of the Fourier Series

f(t) = [t]
-∏<t<∏
T=2∏

Finding a0 and an

Of course, it doesn't have to be t, it could be x or any other variable.

Thank you.

The Attempt at a Solution

You have defined the function as
f(t) = [t]
-∏<t<∏
T=2∏
but that is not the example that you show worked out. The example that you show worked out is this one
f(t) = |t|
-∏<t<∏
T=2∏
This difference being that, in the latter case, the function between - π and +π is the absolute value of t; in the original problem statement, as you gave it, [t] is not really a well defined function.

The absolute value function is an even function, so when we proceed to evaluate the coefficients for the Fourier series, the first one works out as

ao = $\frac{1}{2π}$∫π f(t) dt
= $\frac{1}{π}$∫0π t dt
=$\frac{π}{2}$

The others follow in the usual fashion with the cosine multiplication.

ZedCar said:
I've found an example here;
http://www.exampleproblems.com/wiki/index.php/FS1

Why is it that in my notes 'an' = -4/[∏(k^2)]
That should be either n or k, not both, in your equation. Let's just use n.

What does the result of the integration give you when you assume n is odd and n is even?

The Fourier Series is a mathematical tool used to represent a periodic function as a sum of sine and cosine functions. In this case, the function is f(t) = [t] over the interval -∏<t<∏ with a period of T=2∏. The modulus function, denoted by [t], returns the absolute value of t, meaning that it will always be positive.

To find the Fourier Series of this function, we first need to find the coefficients a0 and an. The coefficient a0 is given by the formula a0 = (1/T) ∫f(t)dt over one period. In this case, we have T=2∏, so a0 = (1/2∏) ∫[t]dt = 1/∏.

The coefficients an are given by the formula an = (2/T) ∫f(t)cos(nωt)dt over one period, where ω = 2∏/T. In this case, we have ω = 2∏/(2∏) = 1, so an = (2/2∏) ∫[t]cos(nt)dt = (1/∏) ∫[t]cos(nt)dt.

To find this integral, we can use integration by parts. Let u = [t] and dv = cos(nt)dt. Then du = sign(t)dt and v = (1/n)sin(nt). By the integration by parts formula, we have ∫uv = uv - ∫vu. Plugging in our values, we get ∫[t]cos(nt)dt = [t](1/n)sin(nt) - ∫(1/n)sin(nt)sign(t)dt. Since the function is periodic, we only need to evaluate this integral over one period, which is from -∏ to ∏.

Using the same substitution, we get ∫(1/n)sin(nt)sign(t)dt = -(1/n^2)cos(nt)sign(t) + ∫(1/n^2)cos(nt)dt. Evaluating this integral over one period, we get ∫(1/n^2)cos(nt)dt = 0, since cos(nt) is an odd function over one period.

Therefore, an = (1/∏) ∫[t]cos(nt

1. What is a Fourier Series?

A Fourier Series is a mathematical tool used to represent a periodic function as a sum of sinusoidal functions. It is named after French mathematician Joseph Fourier and is commonly used in various fields such as signal processing, engineering, and physics.

2. What is the modulus function?

The modulus function, also known as the absolute value function, is a mathematical function that returns the distance of a number from zero on a number line. It is defined as the positive value of a number without regard to its sign.

3. How is Fourier Series of modulus[t] calculated?

To calculate the Fourier Series of modulus[t], the function is first decomposed into its even and odd components. The Fourier Series of each component is then calculated separately and combined to form the final Fourier Series of modulus[t]. This process involves using various mathematical formulas and techniques.

4. What are the applications of Fourier Series of modulus[t]?

The Fourier Series of modulus[t] has various applications, including signal processing, image compression, and harmonic analysis. It is also used to solve differential equations and to study the behavior of periodic systems in physics and engineering.

5. Can Fourier Series of modulus[t] be used for non-periodic functions?

No, Fourier Series is only applicable to periodic functions. If a function is not periodic, its Fourier Series will not exist. However, there are other mathematical tools, such as the Fourier Transform, that can be used to represent non-periodic functions.

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